Problem: A circle has a sector with area $16\pi$ and central angle $\dfrac{8}{9}\pi$ radian. What is the area of the circle? ${36\pi}$ $\color{#9D38BD}{\dfrac{8}{9}\pi}$ ${16\pi}$
Solution: The ratio between the sector's central angle $\theta$ and $2 \pi$ radians is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{2 \pi} = \dfrac{A_s}{A_c}$ $\dfrac{8}{9}\pi \div 2 \pi = 16\pi \div A_c$ $\dfrac{4}{9} = 16\pi \div A_c$ $A_c \times \dfrac{4}{9} = 16\pi$ $A_c = 16\pi \times \dfrac{9}{4}$ $A_c = 36\pi$